View source for Deficiency subspace

<!--
d0306301.png
$#A+1 = 39 n = 0
$#C+1 = 39 : ~/encyclopedia/old_files/data/D030/D.0300630 Deficiency subspace,
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''defect subspace, defective subspace, of an operator''

The orthogonal complement  $  D _  \lambda  $
of the range of values  $  T _  \lambda  = \{ {y = ( A - \lambda I ) x } : {x \in D _ {A} } \} $
of the operator  $  A _  \lambda  = A - \lambda I $, 
where  $  A $
is a linear operator defined on a linear manifold  $  D _ {A} $
of a Hilbert space  $  H $, 
while  $  \lambda $
is a regular value (regular point) of  $  A $. 
Here, a regular value of an operator  $  A $
is understood to be a value of the parameter  $  \lambda $
for which the equation  $  ( A - \lambda I ) x = y $
has a unique solution for any  $  y $
while the operator  $  ( A - \lambda I )  ^ {-} 1 $
is bounded, i.e. the [[Resolvent|resolvent]] of  $  A $
is bounded. As  $  \lambda $
changes, the deficiency subspace  $  D _  \lambda  $
changes as well, but its dimension remains the same for all  $  \lambda $
belonging to a connected component of the open set of all regular values of  $  A $.

If  $  A $
is a symmetric operator with a dense domain of definition  $  D _ {A} $, 
its connected components of regular values will be the upper and the lower half-plane. In this case  $  D _  \lambda  = \{ {x \in D _ {A  ^ {*}  } } : {A  ^ {*} x = \overline \lambda \; x } \} $, 
while the deficiency numbers  $  n _ {+} =  \mathop{\rm dim}  D _ {i} $
and  $  n _ {-} =  \mathop{\rm dim}  D _ {-} i $, 
where  $  A  ^ {*} $
is the adjoint operator, are called the (positive and negative) deficiency indices of the operator  $  A $. 
In addition,

$$ 
D _ {A  ^ {*}  }  =  D _ {A} \oplus D _ {i} \oplus D _ {-} i ,
$$

i.e.  $  D _ {A  ^ {*}  } $
is the direct sum of  $  D _ {A} $, 
$  D _ {i} $
and  $  D _ {-} i $. 
Thus, if  $  n _ {+} = n _ {-} = 0 $, 
the operator  $  A $
is self-adjoint; otherwise the deficiency subspace of a symmetric operator characterizes the extent of its deviation from a self-adjoint operator.

Deficiency subspaces play an important role in constructing the extensions of a symmetric operator to a maximal operator or to a self-adjoint (hyper-maximal) operator.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,   V.I. Sobolev,   "Elements of functional analysis" , Hindushtan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Akhiezer,   I.M. Glazman,   "Theory of linear operators in a Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators" , '''1–2''' , Interscience  (1958–1963)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Riesz,   B. Szökefalvi-Nagy,   "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>

====Comments====
The definition of a regular value of an operator as given above is not quite correct and should read as follows. The value  $  \lambda $
is a regular value of  $  A $
if there exists a positive number  $  k = k ( \lambda ) > 0 $
such that  $  \| ( A - \lambda I ) x \| \geq  k  \| x \| $
for all  $  x \in D _ {A} $. 
In that case the kernel of  $  A - \lambda I $
consists of the zero vector only and the image of  $  A - \lambda I $
is closed (but not necessarily equal to the whole space).